Prove that therea re no solutions in integers x and y to the equation 2x^2 +5y^5=14?

This looks like a job for modular arithmetic! This equation mod 5 says that 2x^2=4. Divide both sides by 2 to get x^2=2 mod 5. But the squares mod 5 are 0^2=0,1^2=1,2^2=4,3^2=4, and 4^2=1. None of these are equal to 2, so x^2=2 mod 5 has no solution.